Second harmonic generation (SHG) is a nonlinear optical process where an optical beam, called the pump beam, interacts with an optically nonlinear medium to generate a second harmonic beam, where the frequency of the second harmonic beam is twice the frequency of the pump beam. Equivalently, the free space wavelength of the second harmonic is half the free space wavelength of the pump. The pump beam can interact with the optically nonlinear medium by passing through the medium and/or by being reflected from the medium. Any material which lacks inversion symmetry can be used as the optically nonlinear medium for SHG. Materials which are commonly used for SHG include LiNbO3 and KTiOPO4 (KTP). For SHG, the nonlinearity of a material is expressed in terms of a second order nonlinear susceptibility tensor χ(2).
Second harmonic generation (especially using a continuous-wave pump) tends to be an inefficient process, where efficiency is the ratio of power emitted in the second harmonic beam divided by the power of the pump beam. The main reason for this inefficiency is that the nonlinearities provided by optically nonlinear materials tend to be weak. Therefore, various measures to improve SHG efficiency have been developed. One way to increase efficiency is to provide more power in the pump beam, because the second harmonic beam power is proportional to the square of the pump beam power in the low efficiency limit (i.e., P2ω<<Pω, where P2ω and Pω are the second harmonic power and pump power respectively). However, the available pump beam power is usually limited, so methods of increasing SHG efficiency for a fixed pump power are of greater interest.
Ensuring phase-matching between the pump beam and the second harmonic beam is the most important of these methods. Phase-matching is collinear if the pump and second harmonic wave vectors are parallel, and non-collinear if the pump and second harmonic wave vectors are not parallel. Non-collinear phase-matching typically leads to the generation of a second harmonic beam which is not parallel to the pump beam. Collinear phase-matching is more commonly employed in practice than non-collinear phase-matching. Assume a pump beam illuminates a section of an optically nonlinear medium. If the phase-matching condition is not satisfied, second harmonic radiation emitted from various points along the illuminated section will interfere destructively, and as a result, the second harmonic beam power will be a periodic function of position, with period 2Lc, along the illuminated section. As taught in U.S. Pat. No. 3,407,309 to R. C. Miller, the coherence length Lc is given by Lc=λ/4Δn, where λ is the free space wavelength of the pump, Δn=|nω−n2ω|, nω is the refractive index of the nonlinear medium at the pump wavelength and n2ω, is the refractive index of the nonlinear medium at the second harmonic wavelength. If the phase-matching condition is exactly satisfied, i.e., nω=n2ω, there will be no destructive interference, and as a result, the second harmonic beam power will increase monotonically along the illuminated section. In a nonlinear device of length L, phase-matching is sufficiently well achieved if Lc is comparable to, or larger than, L. Since L is typically on the order of 1 cm, and λ is typically on the order of 1 μm, Δn must be smaller than about 0.00003 to achieve phase-matching in a typical nonlinear optical device.
Because Δn is typically much larger than 0.00003, due to the dependence of refractive index on wavelength (i.e., dispersion), special methods must be employed to satisfy the phase-matching condition. Two of these methods are birefringent phase-matching (BPM) and quasi-phase-matching (QPM). In a birefringent material, the index of refraction experienced by an optical beam depends on the polarization of the beam. For example, the two states of polarization are called “ordinary” and “extraordinary”, with corresponding indices no and nc, in a uniaxial birefringent medium. BPM is accomplished by selecting a birefringent material which emits second harmonic radiation that is orthogonally polarized to the pump radiation (which imposes certain requirements on the elements of χ(2)) and by ensuring noω≈ne2ω) (or neω≈np2ω). In other words, the difference in index due to dispersion is compensated by the difference in index due to polarization, because the pump and second harmonic beams have different states of polarization.
Birefringent phase-matching is not always possible. For example, a nonlinear material which is not birefringent cannot be birefringently phase-matched. Even for birefringent materials, it is frequently desirable for the polarization of the pump and second harmonic beams to be the same (e.g., to make use of a larger element of the χ(2) tensor, or to avoid the beam walk-off frequently associated with BPM). In these cases, QPM can be employed. As indicated above, in a non-phase-matched interaction, the second harmonic power varies periodically along an illuminated section of nonlinear material with period 2Lc. Let z be position along the illuminated section. The second harmonic power increases to a maximum in the range 0<z<Lc and decreases back to zero in the range Lc<z<2Lc, and this behavior repeats periodically. Thus the contribution of the second coherence length of material to the second harmonic beam exactly cancels the contribution of the first coherence length of material to the second harmonic beam, and the fourth coherence length cancels the third coherence length etc. Basically, the even coherence lengths cancel the odd coherence lengths.
The purpose of QPM is to disrupt this cancellation by periodically altering the properties of a nonlinear material so that each section of length 2Lc makes a net contribution to the second harmonic beam power. This can be accomplished in various ways. One method is to eliminate the nonlinearity of every even coherence length (e.g., by selectively disordering the material to set χ(2) equal to zero). In this case, the even coherence lengths make no contribution to the second harmonic beam, and the above cancellation is eliminated. Another method is to periodically change the sign of χ(2) so that χ(2) in all the even coherence lengths is equal and opposite to χ(2) in all the odd coherence lengths. This periodic alteration of χ(2) can be accomplished by electrical and/or chemical poling of a ferroelectric or other suitable material (e.g., periodic poling of KTiOPO4), or by epitaxial regrowth techniques for semiconductors (e.g., GaAs). The sign change of χ(2) for the even coherence lengths turns destructive interference into constructive interference. In other words, the second harmonic emitted by the even coherence lengths adds constructively to the second harmonic emitted by the odd coherence lengths. Since all parts of the device contribute constructively to the emitted second harmonic when the sign of χ(2) is periodically changed, this form of QPM is preferable to QPM obtained by periodically setting χ(2) to zero.
The above (first order) QPM methods require periodic modification of the properties of a nonlinear material with period 2Lc. Since Lc is typically small (e.g., Δn=0.01 gives Lc=25 μm for λ=1 μm), advanced material fabrication and/or processing technology is typically required for QPM. QPM can also be accomplished by periodically modifying material properties with a longer period (e.g., a period of 6Lc for third order QPM, a period of 10Lc for fifth order QPM etc.), but these higher order QPM methods are less efficient than first order QPM. The purpose of higher order QPM is to disrupt the cancellation of an “odd” section of length nLc by the following “even” section of length nLc, by altering the material properties of each “even” section so that each section of length 2nLc makes a net contribution to the second harmonic beam power. In higher order QPM, n must be odd, so that a section of length nLc makes a nonzero contribution to the second harmonic beam power.
The pump beam for SHG frequently propagates through a nonlinear medium as a Gaussian beam which is brought to a focus (i.e., has a beam waist) inside the nonlinear medium. Phase-matched SHG efficiency increases as the pump intensity and interaction length increase, so it is desirable to maximize both of these parameters. However, increasing the intensity of a beam by bringing it to a smaller focused spot increases beam divergence, which effectively reduces the interaction length. Therefore, there is an optimal waist 1/e amplitude radius w for the pump that maximizes the efficiency of phase-matched SHG in a nonlinear medium of length L. The optimal relation (assuming no beam walkoff between pump and second harmonic) between length L and waist radius w is given by L=Lopt, where Lopt=5.68 πw2nω/λ, and λ is the free space pump wavelength. Since SHG efficiency does not have a sensitive dependence on L for L near Lopt, a nonlinear medium length L in the range of about Lopt/3<L<3 Lopt provides performance that is nearly optimal. The optimal location of the beam waist within the nonlinear medium is at the center of the nonlinear medium (i.e., separated from the entrance and exit faces by a distance L/2).
Other methods of increasing SHG efficiency are frequently employed in addition to phase-matching and optimal focusing. Multipass SHG is one such method, where the pump and second harmonic beams make multiple passes through the nonlinear medium. In multipass SHG, it is necessary to ensure that the pump and second harmonic beams have the proper relative phase in the second and successive passes, so that the contribution of each pass to the second harmonic beam is constructive. J. M. Yarborough et al. (Applied Physics Letters 18(3) 1970) demonstrate double pass SHG in birefringently phase-matched lithium niobate, where a mirror is used to retro-reflect the pump and second harmonic beams through the nonlinear medium, and the separation between the mirror and the crystal is varied to control the relative phase of the two beams in the second pass via the dispersion of air. G. Imeshev et al. (Optics Letters 23(3) 165 1998) demonstrate double pass SHG in quasi-phase-matched lithium niobate, where a mirror is used to retro-reflect the pump and second harmonic beams through the nonlinear medium, and the endface of the nonlinear medium facing the mirror is polished at a small non-zero angle relative to the QPM section boundaries. The relative phase of the pump and second harmonic beams in the second pass is adjusted by translating the nonlinear medium with respect to the beams to vary the medium thickness seen by the beams.
Translating a mirror to control the relative phase of the pump and second harmonic beams on the second pass has the disadvantage that a significant range of motion is required (e.g., on the order of several cm). Translating a wedged nonlinear optical medium to control the relative phase of the pump and second harmonic beams on the second pass is undesirable, because temperature control of the nonlinear medium is typically required, which complicates the design, and the size of the nonlinear medium must be increased to accommodate the translation. Retro-reflection of the pump beam does not preserve optimal focusing of the pump beam from the first pass to the second pass. In other words, if the pump beam is optimally focused for a first pass through a nonlinear medium, and a second pass is obtained by retro-reflection, the second pass pump beam will not be optimally focused through the nonlinear medium.
An object of the invention is to provide improved apparatus and method for adjusting the relative phase of the pump beam and second harmonic beam in multipass SHG. Another object of the invention is to provide apparatus and method for ensuring that the pump beam and second harmonic beam are parallel to each other within the nonlinear medium for all passes. Yet another object of the invention is to preserve optimal focusing of the pump beam for all passes. A further object of the invention is to provide apparatus and method for ensuring that the pump beam and second harmonic beam are parallel to each other within the nonlinear medium for all passes, while also ensuring collinearity of corresponding passes of the pump beam and second harmonic beam (i.e., making the second pass pump beam collinear with the second pass second harmonic beam and making the third pass pump beam collinear with the third pass second harmonic beam etc.).